304
C. E. WATTS
[April
References
1. M. W. Al-Dhahir
and R. N. Shekoury,
Constructions in the hyperbolic plane,
Proc Iraqi Sei. Soc. 2 (1958), 1-6.
2. H. S. M. Coxeter, Non-euclidean geometry, 3d ed., Univ. of Toronto Press, 1957.
3. F. Handest, Constructions in hyperbolic geometry, Cañad. J. Math. 8 (1956),
389-394.
University
of Baghdad
University
and
of Toronto
ON THE EÜLER CHARACTERISTIC OF POLYHEDRA
CHARLES E. WATTS1
1. The principal object of this note is to give an extremely simple
axiomatic characterization
of the Euler characteristic
for finite polyhedra. The idea of the construction used was suggested by the definition of the Grothendieck
group of sheaves over a space [l]. Unless
otherwise noted, "space" always means "triangulable space with base
point," and a "pair" is a "triangulable
pair of spaces with common
base point." Mappings of spaces always preserve base points. If
(X, 4) is a pair, then X/A denotes the space obtained from X by
identifying
4 to a point.
2. Let F be the free abelian group generated by all homeomorphism
classes of spaces. We denote by N the subgroup of F generated by all
elements of the form
X - A - X/A,
where (X, 4) is a pair. Set T = F/N and let 7: F—>Tbe the natural
homomorphism.
Lemma 1. If X, Y are spaces, then y(X\J Y) =yX+yY.
The proof is obvious.
Lemma 2. If En is an n-cell (with base point on the boundary)
yEn = 0.
then
Received by the editors February 27, 1961.
1 This work was done while the author
was a National
Science Foundation
doctoral Fellow.
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Post-
i962]
Proof.
boundary
EULER CHARACTERISTICOF POLYHEDRA
305
Consider the pair (£n+1, £") where £n is imbedded as a
cell. The identification
space being an £n+I, we have
<y.E«+i= <yi<>-|-<yj5>+ii
whence the result.
Lemma
3.
If
Sr
denotes
the
r-sphere,
then
yS" = — 75n+1,
r = 0, 1,2,
Proof.
boundary
Consider the pair (£n+1, 5") where S" is imbedded as the
of £n+1. The identification
space being an 5n+1, we have
rE«+i = 0=7Sn+7.S"+1.
Lemma 4. V is generated by yS".
Proof. Let X be any space, and let £ be a simplicial complex with
|£|
homeomorphic
to X, the base point of X corresponding
to a
vertex of K. If dim K = 0, then X is a wedge of 0-spheres, and Lemma
1 implies that yX is an integral multiple of 75°. Proceeding by induction, suppose that 7 F is an integral multiple of 75° if Y is homeomorphic to a complex of dimension <n, and let X, K be as above
with dim £ = ?7. Let 7 be the (ra-1)-skeleton
of K. Now |£|/|7|
is a wedge of spheres and we have
yA = 77 + 7(| A|/|7|),
so Lemmas
1 and 3 can be applied
to complete
the proof.
Lemma 5. Y is an infinite cyclic group.
Proof. Consider the function defined on the natural basis for F
which assigns to each space its reduced Euler characteristic
(i.e., the
usual Euler characteristic
minus one, or the alternating sum of the
ranks of the reduced
homology groups). This function
extends
uniquely to a homomorphism
£—»Z, vanishing on N, hence induces
a map T—>Z. Since this map assumes the value 1 on 75o and since T
is cyclic by the preceding lemma, the result follows.
Theorem. Let e be any integer-valued function defined for triangulable
spaces with base point, such that
(1) eX = eA+eiX/A)for
(2) eS°=l.
any pair (A, A);
Then e is the reduced Euler characteristic.
Proof. Let x be the function assigning to each space its reduced
Euler characteristic;
x also satisfies (1) and (2). Because of (1), e
and x induce maps ê and x' T—*Z. Because of (2), ex-1 carries the
integer 1 into itself, hence is the identity map. This proves e = x and
« = X-
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306
C. E. WATTS
3. If X, Y are spaces we define the smash product X # Y to be
(XXY)/(X\/Y).
Then the group F above has a ring structure as
the integral algebra of the monoid of all spaces under #. It is easy
to verify that N is a two-sided ideal of F, so that T inherits this ring
structure. As a result, we get an easy proof of the fact that the Euler
characteristic
of the Cartesian product of spaces is the product of
their Euler characteristics.
4. We conclude by suggesting two related problems. First, we remark that the group T has been defined essentially by abstracting
the
definition
of Grothendieck
to a category
by formally replacing
"exact sequence of sheaves" by "simplicial cofibration."
This suggests that we similarly replace it by "simplicial fibration"; i.e., define the subgroup N' of F to be that generated by all elements of the
form E —X—Y, where there exists a fibering of E with fiber Y and
base space X, and then set T' = F/N'. The study of this group V
appears to be quite difficult, and we have as yet no explicit informa-
tion about it.
The second problem
is to define and study other topological invariants in analogous ways. For example, let N" be the subgroup of
the group N above where we allow only those generators corresponding to pairs (X, A) such that 4 is a retract of X. Since the homology
sequence of such a pair splits in each dimension with any coefficient
group, we get in an evident way many nontrivial homomorphisms of
T" = F/N" into the integers by assigning to each space its «th Betti
number mod p, « and p arbitrary. The question arises as to whether
these "Betti number" homomorphisms
determine T" completely, in
the manner in which the "Euler number" homomorphism
determines
T. Again, we have no explicit information.
Reference
1. A. Borel and J.-P. Serre, Le théorème de Riemann-Roch,
86(1958), 97-136.
Institute
for Advanced
Study
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